Third law of thermodynamics

The third law of thermodynamics (or Nernst's theorem after the experimental work of Walther Nernst) states that the entropy of a system approaches a minimum (that of its ground state) as one approaches the temperature of absolute zero. One can write

$\lim _{T\rightarrow 0}{\frac {S(T)}{N}}=0$

where $N$ is the number of particles. Note that there are systems whose ground state entropy is not zero, for example metastable states or glasses, or systems with weakly or non-coupled spins that are not subject to an ordering field.

Implications

The heat capacity (for either pressure or volume) tends to zero as one approaches absolute zero. From

$C_{p,V}(T)=T\left.{\frac {\partial S}{\partial T}}\right\vert _{p,V}$

one has

$S(T)-S(0)=\int _{0}^{x}{\frac {C_{p,V}(T)}{T}}~\mathrm {d} T$

thus $C\rightarrow 0$ as $T\rightarrow 0$, otherwise the integrand would become infinite.

Similarly for the thermal expansion coefficient

$\alpha :={\frac {1}{V}}\left.{\frac {\partial V}{\partial T}}\right\vert _{p}=-{\frac {1}{V}}\left.{\frac {\partial S}{\partial p}}\right\vert _{T}\rightarrow 0$

References

1. P. T. Landsberg "A comment on Nernst's theorem", Journal of Physics A: Mathematical and General 22 pp. 139-141 (1989)